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Translation equivalent elements in free groups. (English) Zbl 1112.20023
Two elements \(g,h\) of a free group \(F_n\) of rank \(n\) are called translation equivalent if the cyclic length of \(\varphi(g)\) equals the cyclic length of \(\varphi(h)\) for every automorphism \(\varphi\) of \(F_n\). In this paper, the author proves the following Theorem: Let \(g,h\in F_n\) be translation equivalent in \(F_n\) and let \(w(a,b)\in F(a,b)\) be arbitrary. Then \(w(g,h)\) and \(w(h,g)\) are translation equivalent in \(F_n\). Here \(F(a,b)\) is the free group on \(a,b\). This theorem answers in the affirmative Problem F38b in the online version [http://www.grouptheory.info] of G. Baumslag, A. G. Myasnikov and V. Shpilrain, Open problems in combinatorial group theory. Second ed. [Combinatorial and geometric group theory. Contemp. Math. 296, 1-38 (2002; Zbl 1065.20042)]. The proof here is combinatorial.

MSC:
20E05 Free nonabelian groups
20E36 Automorphisms of infinite groups
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
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[1] DOI: 10.1023/B:GEOM.0000006579.44245.92 · Zbl 1037.57013 · doi:10.1023/B:GEOM.0000006579.44245.92
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